Download THE PROOF IS IN THE PICTURE Theorem: The square root of 2 is

THE PROOF IS IN THE PICTURE
(an introduction to proofs without words)
LAMC INTERMEDIATE GROUP - 11/24/13
WARM U P (PAPER F OLDING )
Theorem: The square root of 2 is irrational.
We prove this via the mathematical technique of contradiction. To that end, suppose
that the square root of 2 was rational. This means that we can find a right isosceles
triangle with integer sides.
Follow these steps to prove this is impossible:
• Take out a square piece of paper. Suppose your square has side-length n units
and that the diagonal has length m units (n and m integers).
• Fold it in half along the diagonal so that you have a right isosceles triangle.
• Fold a leg of your triangle onto the diagonal. (See the figure below)
• Once you have finished folding your triangle, flip to the next page to work out
the details.
This proof originally by John Conway
Copyright c 2008-2013 Olga Radko, Deven Ware/Los Angeles Math Circle/UCLA Department of Mathematics .
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Recall the following facts:
• The sum of the degrees of the angles in a triangle is 180.
• In a right triangle the side lengths are related to the angles. In particular if
the two non-right angles are the same you have an isosceles triangle.
Use these facts to fill in the missing lengths: CF, FA, EF, EB and EA in the following
picture.
You have now shown that triangle AFE is a right isosceles triangle. Find the contradiction.
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P ROOFS
BY
P UZZLE
For each problem in this section, you are given a theorem and puzzle pieces. The
goal is to arrange the puzzle pieces in some fashion to prove the theorem. Here is
an example (my picture has no explanation, but please explain why your pictures are
proofs when you solve the subsequent problems!)
• Question: Prove that 32 + 42 = 52 using the following puzzle pieces:
• Solution:
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(1) The triangular number tn is defined as
tn = 1 + 2 + · · · + n.
Prove that tn = n(n + 1)/2 given the following puzzle pieces:
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(2) Prove that the sum of the first n odd numbers 1 + 3 + · · · + (2n
the following puzzle pieces:
1) = n2 using
Draw separate pictures for n = 2, n = 3, and n = 4 then explain how exactly
to generalize for any number n.
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(3) In this problem, we prove the same theorem as in problem 2 using different
puzzle pieces. Prove that the sum of the first n odd numbers
1 + 3 + · · · + (2n
using the following puzzle pieces:
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1) = n2
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(4) In this problem you will prove two related identities.
• Prove that 3tn + tn
1
= t2n using the following puzzle pieces:
• Prove that 3tn + tn+1 = t2n+1 using the following puzzle pieces:
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(5) (Challenge) The goal is to prove the identity t2(n 1) + t2n = tn2 .
• Prove that t21 + t22 = t22 using the following puzzle pieces:
• Find your own puzzle pieces to prove that t22 + t23 = t32 = t9 and explain how
to generalize this to any number n.
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(6) Prove the Pythagorean theorem using the following puzzle pieces:
Hint: You should draw a big square (with side-length a + b) in two different ways
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(7) Prove that F02 + F12 + · · · + Fn2 = Fn Fn+1 where Fn is the nth Fibonacci number
using the following puzzle pieces:
Draw separate pictures for n = 1, n = 2, n = 3 and n = 4 then explain how
exactly to generalize for any number n.
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E VALUATE
THE INFINITE SUMS !
(1) Evaluate the infinite sum
1
2
+ 14 + 18 +
(2) Evaluate the infinite sum
1
4
+
1
16
+
1
64
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1
16
+ · · · using the following picture:
+ · · · using the following picture:
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P ICTURE P ROOF J EOPARDY
In this section, we present a picture proof but don’t tell you the theorem it is proving.
See if you can figure out what is going on in the picture and provide the theorem it
proves.
(1) Hints: Figure out the length and width of this rectangle. Now you can write
the area as either length times width or as the sum of the areas of the inside
pieces.
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(2) Hint: This is about a correspondence between two things you can count.
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(3) (Challenge) This is another proof of the irrationality of the square root of two
(due to Tennenbaum). Try to explain the picture.
p
p
Hint: Start with 2 = ab with a and b as small as possible. Note that 2 > 1
so we have a > b. Now the p
picture makes sense, try to use it to find a smaller
fraction representation for 2.
Math Circle (Intermediate) 11/24/2013
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